since the ring $R$ is Boolean, it is unitary, commutative and has no non-zero nilpotent element. hence $R$ is the direct sum of $n$ copies of $\mathbb{F}_2.$ now for any $1 \leq i \leq n$ let $a_i=(x_{i1}, \cdots , x_{in}) \in R,$
where $x_{ii}=1$ and $x_{ik}=0, \ \forall k \neq i.$ obviously $a_ia_j=0, \ \forall i \neq j. \ \ \Box$